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La Cañada Math II Advanced Newsletter Unit 1 – Geometric Foundations and Reasoning Unit 1 Overview In this introductory unit, students will be exposed to the language that forms the foundation of geometry, such as point, line, plane, segment, ray, distance, angle, etc. The unit will open with an introduction to several important undefined terms; point, line and plane, and how those terms are illustrated geometrically and with appropriate notation. Students will use Geometer’s Sketchpad to solidify their understanding of these terms and familiarize themselves with the protocols for using geometric software. As an additional representation, points and line segments will be explored on the coordinate plane, and formulas will be developed to measure the distance between two points and the midpoint of the segment joining two points. Students will then be introduced to the definition of an angle, and how it can be defined using rays with a common endpoint or in terms of the arc length of a circle. The definitions discussed in the first part of this unit will lead to several postulates that students will use to describe the characteristics of a diagram. An additional emphasis will be on construction, which is the ability to create a precise diagram with a straightedge and compass, and no need for rulers, protractors, or the approximation that they carry with them. Students will relate each construction to the definition of the term and use them throughout the course as a connection to axiomatic logic and proof. Once the appropriate definitions are in place, the focus of the unit will shift to the use of inductive and deductive reasoning in mathematical sense-making. Students will learn how inductive reasoning allows for conjectures to be formed out of pattern recognition and that such a conjecture can only be considered valid if it is proven deductively. Students will use properties of equality and congruence, along with previous and new definitions and postulated to prove conjectures. These proofs will require an understanding of an organizational system (a 2-column proof at this introductory stage) and logic, in order to prove the assumption laid out in the conjecture. Proofs will include the Midpoint Theorem, Angle Bisector Theorem, definition of vertical angles, and other conjectures involving complementary and supplementary angles. Important Dates August 31 Unit 1 Test (Tentative) Homework Policy Daily Homework will be assigned and should be completed prior to the next class meeting. For more details about assignments, check the class website. Weekly Homework will be a component of the course, and will emphasize algebra skills learned in last year’s course in an attempt to maintain important ideas before entering LC Math III. Technology Note We will be using various geometry software programs this year in class. Geometer’s Sketchpad is a program available online, but only in a trial capacity. In school, we will visit the Mac Lab to use this program. Geogebra is a free online program that will also be useful at home and in class. If you are absent from a class period in which these programs are used, feel free to reach out to the teacher to insure that the assignment can be completed on a school computer or at home using the appropriate technology. Page 1 of 2 Additional Unit 1 Information Key Common Core Standards G.CO.1 - Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. G.CO.9 - Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. G.CO.12 - Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 8.G.8b - Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. G.GPE.6 - Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Helpful Resources Geometry Textbook – Chapters 1 and 2, Chapter 13 (Sections 1 and 5 only) Construction Support: www.mathsisfun.com/geometry/constructions.html Geogebra: www.geogebra.org Dr. Carruthers ([email protected]) Mr. McDermott ([email protected]) Spotlight on the Standards for Mathematical Practice: MP 3: Construct viable arguments and critique the reasoning of others. Constructing arguments is an important mathematical practice. Mathematics, as a field of study, is characterized by a need to prove what is true. As students advance in their mathematics education, they will be asked to justify and prove mathematical relationships with an increasing level of rigor and formality. This unit will introduce students to the idea of proof, and making a logical argument that is supported by definitions and postulated. Deductive reasoning will be a skill that will take time to develop throughout the course, but is a critical byproduct of the geometry knowledge gained through this course. Students will practice sequencing statements logically and supporting claims with evidence, both important skills that are transferrable to other disciplines. Page 2 of 2